Build medium-hot fire in charcoal grill, or heat gas grill to medium. Place chicken on greased grill grate and cook for 7 minutes per side, or until thickest part of the meat reaches internal temperature of 165 F. Let chicken rest for 10 minutes; slice into 1 inch pieces.

In blender container, place 3 cups watermelon and blend until smooth. Press through fine-mesh strainer into bowl; discard pulp. Pour 1 1/2 cups watermelon juice into small saucepan. Add shallots, chili sauce, soy sauce, fish sauce, lime juice, and gingerroot to saucepan. Bring to a boil; reduce heat to medium and cook, uncovered, until liquid is reduced by about one half, 20 to 25 minutes. Stir in sesame oil. Remove from heat and let cool.

Meanwhile, prepare noodles according to package directions; rinse with cold water and drain well.

In large bowl, place noodles. Add cooked chicken pieces, carrots, radish, cilantro, mint, and remaining watermelon. Drizzle with watermelon sauce; toss to coat.

Divide among serving plates and top with green onions and peanuts. Serve with lime wedges.